Re: Why does everyone do it?

On 31 Aug, 05:26, Virgil <Vir...@xxxxxxxxx> wrote:
In article
<5fbce152-bcca-438c-a68e-d7a808fcc...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
 ju...@xxxxxxxxxxxxx wrote:
On 31 Aug, 03:13, Keith Ramsay <kram...@xxxxxxx> wrote:
On Aug 27, 11:52 am, herbzet <herb...@xxxxxxxxx> wrote:

|So the diagonal argument does not establish the uncountability
|of the set of provably well-defined sequences, just the non-
|enumerabiliy of that set.

What's interesting (not mathematically, but psychologically)
is how people get the feeling that "uncountability" somehow
means something different from what you're calling
"non-enumerability". Quite a lot of the troubles people have
with the diagonal argument seem to me to result from not
quite believing that the term "uncountable" means just what
it's defined to mean, and not some subtly different and
stronger thing.

|So, if the reals are likewise established as non-enumerable
|by a diagonal argument, we likewise need not conclude that they
|are also uncountable, but merely that the reals are not (recursively)
|enumerable.

It doesn't make sense to add the qualifier "recursively"
here. They're not enumerable by whatever kind of enumeration
you consider coherent.

I'm with you up to this pont. Then I do not undertand.

We do not understand "pont". Do you mean "point"?

What do you think?

A constructivist might believe that
only computable enumerations exist, and only computable reals,
but (using the same argument) there is no computable
enumeration of the computable reals, so they would consider
the reals uncountable.

Would a constructivist accept *that* argument in the very first place?

Whyever not?

At least by the  definition given in
   http://en.wikipedia.org/wiki/Constructivism_(mathematics)

"This then opens the question as to what sort of function from a
countable set to a countable set, such as f and g above, can actually
be constructed. Different versions of constructivism diverge on this
point. Constructions can be defined as broadly as free choice
sequences, which is the intuitionistic view, or as narrowly as
algorithms (or more technically, the computable functions), or even
left unspecified. If, for instance, the algorithmic view is taken,
then the reals as constructed here are essentially what classically
would be called the computable numbers."

That page just reflects the overall confusion on constructivism. I was
hoping for some hints *about* this. I know about Wikipedia. Though
thanks.

Is there maybe a specific kind of constructivism (or constructivists)
you are refering to here?

(For instance, to me the connection between intuitionism and
constructivism *in mathematics* is far from clear. Who is who?)

Sorry, I might be asking a lot. Maybe few names or links might do.

Try doing some of your own research.

And then there still is that thing of the impredicativity. I would
think a constructivist -- in mathematics -- wouldn't accept anything
like that, would s/he?

I am truly not sure about this, given that even such a basic thing as
the proof for the irrationality of sqrt(2) seems to rely on similar
kinds of principles, like assuming the real numbers in the premises.

I mean, I know such an approach is possible when one starts the number
hierarchy from the reals. But I can't see how to go in the opposite
direction: from the counting numbers to the rationals is easy, but how
to get the irrationals *constructively* (without "impredicativity in
the premises")?

-LV
.

Relevant Pages

  • Re: Why does everyone do it?
    ... is how people get the feeling that "uncountability" somehow ... but merely that the reals are not ... They're not enumerable by whatever kind of enumeration ... only computable enumerations exist, and only computable reals, ...
    (sci.math)
  • Re: Why?
    ... that the reals are uncountable is both classically and constructively ... Without a precise definition, ... uncountability of the reals would one claim not to be ... To quote from the Wikipedia article on constructivism: ...
    (sci.logic)
  • Re: Why does everyone do it?
    ... is how people get the feeling that "uncountability" somehow ... but merely that the reals are not ... They're not enumerable by whatever kind of enumeration ... Is there maybe a specific kind of constructivism ...
    (sci.math)
  • Re: Why does everyone do it?
    ... is how people get the feeling that "uncountability" somehow ... but merely that the reals are not ... They're not enumerable by whatever kind of enumeration ... Is there maybe a specific kind of constructivism ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > extended reals, of course, instead of the real numbers, or by ... point for the enumeration of the finite reals, if that makes you feel better. ... strings is always longer than it is wide. ... "uncountability" of the real numbers, and the impossibility of a well ordering, ...
    (sci.math)